## Abstract

A new kind of partially coherent beam with non-conventional correlation function named generalized multi-Gaussian correlated Schell-model (GMGCSM) beam is proposed. The GMGCSM beam of the first or second kind is capable of producing dark hollow or flat-topped beam profile in the focal plane (or in the far field). Furthermore, we carry out experimental generation of a GMGCSM beam of the first or second kind, and measure its focused intensity. Our experimental results verify theoretical predictions. The GMGCSM beam will be useful for free-space optical communications, material thermal processing, particle or atom trapping.

© 2014 Optical Society of America

## 1. Introduction

Since Gori et al. discussed the sufficient conditions for devising genuine correlation functions of scalar and electromagnetic partially coherent beams [1, 2], a variety of partially coherent beams with non-conventional correlation functions have been introduced [3–21]. Partially coherent beams with non-conventional correlation functions display many extraordinary properties, such as far-field flat-topped [4–7], four-beamlet array [8] and dark hollow (i.e., ring-shaped) beam profile formation [9–12], optical cage formation near the focal plane [13], self-focusing and a lateral shift of the intensity maximum [14, 15], far field radially polarization formation [16] and are useful in free-space optical communications, material thermal processing and particle trapping [13, 17–21]. Multi-Gaussian correlated Schell-model (MGCSM) beam (i.e., multi-Gaussian Schell-model beam) whose degree of coherence (i.e., correlation function) is modeled by multi-Gaussian distribution was proposed by Korotkova et al. [4–6], and such beam has a flat-topped beam profile in the far field (or in the focal plane). Propagation properties of various MGCSM beams have been studied in detail [4–7, 20, 21], and it was found that MGCSM beams are less affected by turbulence than conventional Gaussian Schell-model beams and are useful in free-space optical communications. In this paper, a generalized multi-Gaussian correlated Schell-model (GMGCSM) beam whose degree of coherence is also modeled by multi-Gaussian distribution is proposed. The proposed GMGCSM beam of the first or second kind is capable of producing dark hollow or flat-topped beam profile in the focal plane (or in the far field). Experimental generation of a GMGCSM beam of the first or second kind is reported.

## 2. Generalized multi-Gaussian correlated Schell-model beam: Theory

The statistical properties of a scalar partially coherent beam are characterized by the mutual coherence function (MCF) in the space-time domain [22]. The MCF of the GMGCSM beam is defined as

*N*is the beam index. For the case of$\alpha =D$, ${A}_{mn}^{\alpha},{B}_{mn}^{\alpha},{C}_{\alpha}$ take the following form

*N*increases.

In general, the GMGCSM beam can’t be used to describe the MGCSM beam by choosing suitable beam parameters, while the distribution of the DOC of the GMGCSM beam of the second kind is similar to that of the DOC of the MGCSM beam [4, 5]. The MGCSM beam is capable of producing flat-topped beam profile in the far field (or in the focal plane), while the GMGCSM beam of the first or second kind is capable of producing dark hollow or flat-topped beam profile in the far field (or in the focal plane). In this way, the GMGCSM beam is more general.

The MCF of the partially coherent beam should satisfy the condition of nonnegative definiteness [1, 22] and can be written in the form [1]

*H*is an arbitrary kernel,

*I*is a non-negative function and the asterisk denotes the complex conjugate. Equation (7) can be expressed in the following alternative form

*I*and

*H*. Here

*I*and

*H*represent the intensity of the incoherent source and the response function of the optical path, respectively.

If we set *H* and *I* as follows:

*H*in the form of Eq. (10) and

*I*as follows:

Applying Eqs. (1), (2) and the generalized Collins formula [23, 24], we obtain the following expression for the MCF of a GMGCSM beam in the output plane after propagating through a stigmatic ABCD optical system

*A*,

*B*,

*C*and

*D*are the transfer matrix elements of the optical system; $k=2\pi /\lambda $ is the wave number with $\lambda $ being the wavelength. The intensity of the output GMGCSM beam is obtained as$I\left(\rho \right)=J(\rho ,\rho )$.

As a numerical example, we study the propagation properties the GMGCSM beam of the first or second kind in free space. The elements of the transfer matrix for free space of distance z read as *A* = 1, *B* = z, *C* = 0 and *D* = 1. Figure 2 shows the normalized intensity distribution (cross line ${\rho}_{y}=0$) of the GMGCSM beam of the first kind (Figs. 2(a)-2(c)) or the second kind (Figs. 2(d)-2(f)) at several propagation distances in free space for different values of beam index *N* with ${\sigma}_{0}=1\text{mm}$, ${\delta}_{0}=0.2\text{mm}$ and $\lambda =632.8\text{mm}$. One sees clearly that the intensity distribution of the GMGCSM beam of the first or second kind in the source (z = 0) has a Gaussian profile and is independent of the DOC (i.e., correlation function) and the beam index *N*. With the increase of the propagation distance, the effect of DOC on the intensity distribution of the GMGCSM beam gradually appears. The intensity distribution of the GMGCSM beam of the first or second kind turns to dark hollow (see Fig. 2(c)) or flat-topped (see Fig. 2(f)) profile in the far field, and the dark size or flatness of the beam spot increases as the beam index *N* increases.

## 3. Generalized multi-Gaussian correlated Schell-model beam: Experiment

Now we carry out experimental generation of the proposed GMGCSM beam of the first or second kind with *N* = 1. The most important part in our experiment is to construct an incoherent source whose MCF is given by Eq. (9) and intensity is given by Eq. (11) or Eq. (12). Part I of Fig. 3 shows our experimental setup for generating the GMGCSM beam of the first or second kind with *N* = 1. A linearly polarized Gaussian beam emitted from a He-Ne laser beam ($\lambda \approx 633\text{nm}$) first passes through a half-wave plate (${\text{HP}}_{1}$), then passes through a Mach-Zehnder interferometer (MZI). Two orthogonally polarized beams are superimposed together at the output of the MZI. Here ${\text{HP}}_{1}$ is used to control the amplitudes of the *x* and *y* linearly polarized beams through rotating the ${\text{HP}}_{1}$. The beam expander (BE) in MZI is used to control the width of the *x* linearly polarized beam. The electric field of the beam at the output of the MZI can be expressed as

*x*- and

*y*-directions, respectively, ${a}_{x}$and ${a}_{y}$denote the amplitudes of the

*x*and

*y*components of the field. ${\omega}_{0}$and ${\omega}_{1}$denote the beam width of the

*x*and

*y*components of the field. In our experiment, we set ${\omega}_{1}=\sqrt{2}{\omega}_{0}$through varying the BE, and we set ${a}_{x}={a}_{y}$through varying the${\text{HP}}_{1}$. The output beam from the MZI passes through a linearly polarizer (LP) whose transmission axis forms an angle

*θ*with the

*x*axis and a ${\text{HP}}_{2}$which is used to rotate the polarization direction of the beam along

*x*-direction. Then the intensity distribution of the beam from the ${\text{HP}}_{2}$is given by

If the transmission angle *θ* of LP equals to$-\pi /4$, Eq. (17) reduces to

*N*= 1 except for the coefficient${a}_{x}^{2}/2$, which is not an important coefficient. If

*θ*equals to$-0.35\pi $, Eq. (17) reduces to

*N*= 1 except for the coefficient$5{a}_{x}^{2}/4$. The transmitted beam from the ${\text{HP}}_{2}$illuminates on a rotating ground-glass disk (RGGD), producing a partially coherent source with Gaussian statistics. If the diameter of the beam spot on the RGGD is much larger than the inhomogeneity scale of the ground glass [25] which is satisfied in our experiment, the generated partially coherent source can be regarded as an incoherent source whose MCF is given by Eq. (9) and intensity is given by Eq. (18) for $\theta =-\pi /4$ or Eq. (19) for $\theta =0.65\pi $. After passing through the thin lens L

_{1}and the GAF with $T\left(r\right)=\mathrm{exp}\left(-{r}^{2}/4{\sigma}_{0}^{2}\right),$the incoherent beam with prescribed intensity becomes a GMGCSM beam of the first kind for the case of $\theta =-\pi /4$ or the second kind for the case of $\theta =0.65\pi $ with

*N*= 1.

Part II of Fig. 3 shows our experimental setup for measuring the square of the modulus of the DOC and the focused intensity of the generated GMGCSM beam. The generated beam from the GAF is split into two beams by the BS. The reflected beam passes through the thin lens L_{2} with focal length *f*_{2}, and arrives at the CCD. Both distances from GAF to L_{2} and from L_{2} to CCD are 2*f*_{2} (i.e., 2 *f*-imaging system), so the DOC of the beam in the CCD plane is the same as that in the source plane (just behind the GAF). The output signal from the CCD is sent to a PC to measure the square of the modulus of the DOC. The detailed principle and the measuring process can be found in [3].

The transmitted beam from the BS passes through the thin lens L_{3} with focal length *f*_{3} = 10cm, then arrives at the BPA, which is used measure the intensity distribution. The distances from GAF to L_{3} and from L_{3} to BPA are *f*_{3} and *z*, respectively. The elements of the transfer matrix between GAF and BPA read as

Figure 4 shows our experimental results of the square of the modulus of the DOC of (a) the generated GMGCSM beam of the first kind and (b) the generated GMGCSM beam of the second kind with *N* = 1. Through theoretical fit of the experimental data, we obtain ${\delta}_{0}=0.054\text{mm}$for the generated GMGCSM beam of the first kind and ${\delta}_{0}=0.035\text{mm}$for the generated GMGCSM beam of the second kind. One finds that the distribution of the square of the modulus of the DOC of the generated GMGCSM beam of the second kind is quite similar to Gaussian distribution, which is caused by the fact that the side lobe is too weak to be detected in our experiment, while the focusing properties of the generated GMGCSM beam is not influenced.

Figures 5 and 6 show our experimental results of the intensity distribution and the corresponding cross line (dotted curve) of the generated GMGCSM beam of the first kind and the generated GMGCSM beam of the second kind in the source plane and at two propagation distances after passing through the thin lens L_{3}. For the convenience of comparison, the theoretical results calculated by Eq. (13) are also shown in Figs. 5 and 6. One finds from Figs. 5(a) and 5(d), Figs. 6(a) and 6(d) that the generated GMGCSM beam of the first kind has the same intensity distribution with that of the generated GMGCSM beam of the second kind in the source plane with ${\sigma}_{0}=1\text{mm}$. From Figs. 5(b), 5(e), 5(c), 5(f), Figs. 6(b), 6(e), 6(c) and 6(f), one sees that the generated GMGCSM beam of the first kind and the generated GMGCSM beam of the second kind display quite different focusing properties, the former produces dark hollow beam profile while the latter produces flat-topped beam profile in the focal plane (or in the far field) due to different initial correlation functions. The focusing properties GMGCSM beam of the second kind is similar to that of the MGCSM beam proposed in [3, 4] which also displays flat-topped beam profile in the focal plane (or in the far field). Our experimental results are quite consistent with the theoretical predictions.

The experimental setup in this paper is only for generating GMGCSM beams with *N* = 1. In principle, the GMGCSM beams with higher values of *N* also can be generated for modifying the Mach-Zehnder interferometer in the experimental setup. Generation of the GMGCSM beams with *N* = 1 requires superposition of two Gaussian fields, while generation of the GMGCSM beams with *N*>1 requires superposition of 2*N* Gaussian fields in the interferometer. The interferometer should be carefully devised to realize the GMGCSM beam with higher values of *N*, and we leave this for future study.

## 4. Summary

We have introduced a new kind of partially coherent beam with non-conventional correlation function named GMGCSM beam, and derived its paraxial propagation formula. We have found that the GMGCSM beam of the first kind produces dark hollow beam profile and the GMGCSM beam of the second kind produces flat-topped beam profile in the focal plane (or in the far field). Furthermore, we have generated the GMGCSM beam of the first or second kind, and studied its focusing properties experimentally. Our experimental results have verified theoretical predictions. The proposed GMGCSM beam of the first kind will be useful for particle or atom trapping, where dark hollow beam spot is used to trap a particle [26] or atom [27]. The GMGCSM beam of the second kind will be useful in material thermal processing and free-space optical communications, where flat-topped beam spot is preferred [20, 21, 28].

## Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11474213&11404007&11274005&11104195, the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Universities Natural Science Research Project of Jiangsu Provsince under Grant No. 11KJB140007, Anhui Provincial Natural Science Foundation of China under Grant No.1408085QF112, and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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